Restablishing mi blog!

I just found an oeld artikl that I had riten in simplified speling a whiel bak. It deels with th subjekt of mathematiks reform, and I am hiely pleezd with it. I'm going to paest it "as is" for th tiem being. Laeter I mae poest a vurzhon in "traditional" orthografy. I shuud point out that th artikl is riten as if it cuud be th script for a raedio or TV proegram, so thaer ar sum referenses to "lafter". This wuud be lafter from a prezuemd "audi'enss". Utherwiez, th text of th artikl shuud be self-explanitory...

Whi Jony haets math

With this artikl I am inaugeraeting a nue seerys of artikls on “mathematikal sanity” that wil ultimatly sho whi Jony haets math and whot can be dun about it.

To understand whaer th problem starts let’s furst go all th wae bak to furst graed and see how poor Jony was taut to count. At furst things did not luuk so bad. He was taut to count bi wuns, which is how wun uezhualy starts lurning to count. This leeds to th familiar set-up with th Hindoo-Aerabik noomerals being listed in order as foloes: 0, 1, 2, 3, 4, 5,6, 7, 8, 9, 10, 11, 12, 13… and so on.

It is ofen nesesaery, however, to count bi 2’s, bi th eeven numbers. Jony has meerly to luuk around his hoem, outsied in th yard, or in his clasroom, and he wil be surpriezd bi th number of things that cum in paers. Ietems of cloething such as shoes ar moest redily noetist, but Jony soon discuvers that eeven when ietems do not cum in paers but ar lieing about randomly it maeks for faster counting if wun can count them bi toos insted of wuns. Forchunatly, this is not too hard sinss th eeven numbers ar faerly prediktabl. Thus we hav th familyar set of noomerals 0, 2, 4, 6, 8, 10, 12, 14, ets. Thus so far everything seems to be oekae. After reeching ten th patern of eeven numbers simply repeets itself all oever agen, and all Jony has to do to tel whether a number is eeven or od is to meerly luuk at th dijit in th “wuns” plaess, and he in no daenjer of loozing count no mater how larj th numbers get.

But whot hapens if Jony wonts to or needs to count bi 3’s? Now, wun miet think that being aebl to count bi threes wuud not be all that important. However, if Jony luuks around, he wil discuver that a graet meny things do in fakt cum in threes. It mae simply be a roe of flag-stoens three colems wied, or a roe of seets in a moovy theeatr, or a seerys of telefoen poels that cum in sets of three as far as th ie can see. Now, it wuud be niess if Jony cuud aulso count thees ietems as eezily as he cuud when he was counting bi wuns or bi toos.  Unforchunatly, however, this is not to be th caess. When Jony tries to count bi threes sumthing verry straenj happens. At furst things luuk OK, but then, wel, see for yuurself!:

U get 0, 3, 6, 9, 12, 15, 18, 21, 24 and so on!!!

Poor Jony fiends oenly confuezhon. After going past ten th siekl of numbers duz not repeet itself in eny simpl wae. If fakt, if he went far enuf Jony wuud evenchualy discuver that a multipl of three can end in eny of th ten posibl Hindoo-Aerabik noomerals. Thaer is no eezy wae to tel whether a number is a multipl of three or not! Jony’s exporaeshon of mathematikal reality is begining to faulter. Chanses ar, if he is a tipikal chield of an aej to be in th furst graed or so, he wil evenchualy looz count and then compleetly looz interest in trieing to count objekts that cum in threes.

But Jony’s problems ar oenly just beginning; sooner or laeter he is going to faess th task of counting bi 4’s. Now, if wun stops to think about it, praktikaly everything seems to cum bi fors: Moest taebls hav for legs, moest cars and uther veeikls hav for wheels, and, of corss, moest “for-leged” animals aulso hav for legs. Whot’s mor, moest windows and dors hav for sieds—tho thae  mae not all be of th saem siez—and moest ruums and moest houses aulso hav for wauls. U miet eeven sae that our entier siviliezd wurld consists mor or les of for-sieded objekts!! But luuk at whot happens when Jony tries to count bi fors:

 0, 4, 8, 12, 16, 20, 24...

 To be shuur, at th verry leest th numbers ar all eeven, which duz help a bit, but for such an elementaery counting task th desimal sistem needlesly complicaets Jony’s atempts at lurning to count bi fors. This is becauz th number 4 duz not divied eevenly into ten, with th rezult that th siekl of  wun-dijit endings, -0, -4, -8, -2, -6, ets., duz not repeet itself in a simpl straetforward fashon. Sinss a multipl of for can end in eny eeven number in baess ten thaer is no eezy wae for a smaul chield hoo is lurning how to count to tel whether a number is a multipl of for or not—and wun can be shuur that th teecher in a “desimal” clasroom is definitly not going to stres counting bi fors, so Jony wil not get much help from skool eether!).

But now heer’s sumthing straenj!:  Jony wil evenchualy noetis that counting bi fievs is parrodoxikaly simpl; when wun counts repeetedly bi fievs in baess ten wun gets th foloeing simpl patern: 0, 5, 10, 15, 20, 25, 30, and so on. It is eezy for eny kid to see heer that a multipl of fiev can oenly end in zeero or fiev. As a mater of fakt, it is probaly so obvius and saeli’ent a feecher of th desimal sistem that chanses ar no teecher wuud ever need to point it out to th clas (tho moest do go out of thaer wae to point it out enywae, if mi memory of mi oen skool daes survs me wel!!). But at this point, however, we need to ask ourselvs a critikal qeschon: Whot guud duz this “miracuelus ability” to count bi fievs reealy do for Jony? It turns out that out of all th uther numbers we hav explord so far, it is hard to think of enything in th “reeal” wurld that cums bi fievs; thaer ar vurchualy no fiev-leged taebls—exsept for broeken wuns, perhaps, no fiev-wheeld veeikls, no fiev-leged animals, no fiev-sieded windoes or dors, no fiev-sieded ruums, boxes or cuebs. It is troo, wun miet argue, that Jony can aulwaes count on his finggers. So, oekae, Jony duz hav for finggers and a thum on eech hand, asueming that he is normal for his aej…But whi, wun mae ask, duz Jony reealy need to count his finggers all th tiem? Now maebe if Jony had a pet aligaetor or a pet piraana at hoem he miet reealy need to count his finggers every morning to maek shuur thae wer stil thaer! (lafter). Utherwiez, however, U wuud think that he wuud get bord of counting his finggers all th tiem!!...

Nacheraly, of corss, we can oenly taek this joek so far!; th iedeea is, in prinsipal, that Jony is supoest to go around luuking at objekts in th reeal wurld and then to tri maching them up with th ten fingers and thums that he has on boeth hands. This is not an efishent method of counting, however. If a faktory wurker needs to count objekts that ar mooving rapidly doun a convaeor belt—espeshaly if thae ar cuming in baches of too, three, or for, which is qiet ofen th caess—he or she wil hav to luuk direkly at th objekts and count them. Thaer wil simply be no tiem for luuking bak and forth at wun’s finggers and then at th objekts, and th numbers that ar counted wuud raerly mach up with fiev or ten or multipls of thees numbers in eny event. Ultimatly, Jony needs to lurn to count objekts in th reeal wurld instead of relieing on th cruch of primitiv fingger counting. And fienaly, if Jony reealy duz need to relie on parts of his body for counting, thaer ar plenty of diferent waes that he can uez them to count bi toos, threes, or fors, and thees waes ar all far mor liekly to mach up with th objekts that he is akchualy counting!

So thaer U hav it: Counting bi toos, threes, and fors is verry uesful, espeshaly sinss it is ofen posibl to imeediatly recogniez a groop of too, three, or eeven for objekts at a glanss without having to deliberatly count them. Bi th tiem we get to groops of objekts that cum in clusters of fiev or mor, however, it is ofen dificult to imeediatly tel how meny objekts we ar deeling with unles we deliberatly stop and count them, and this in turn involvs counting them eether indivijualy or bi cleerly vizibl groopings such as toos, threes, and so on.

Admitedly, however, we can cheet a bit, espeshaly if we ar deeling with objekts that ar araed in roes as wel as in colums. Taek, for exampl, th familyar “six-pak” of beverajes or “haf-duzen” of egs. Now, if thees objekts ar araed in a 2 bi 3 patern wun can eezily see that counting bi sixes is eqivalent to counting bi toos three tiems or to counting bi threes twiess. Thus we can aulredy see that counting bi sixes is mor important than counting bi fievs. Whot’s mor, if Jony meerly luuks around him, he wil noetis that qiet a number of objekts cum in sixes or relaet to th number six in sum wae. Reealy long taebls mae hav six legs, a lot of delivery truks hav six wheels, and whot litl kid duzn’t noe that moest bugs—yuk!—hav six legs apeess (at leest when thae’r hoel…lafter). Then after that, thaer is every chield’s faevorit toy, a bag of bloks, and every blok aulwaes has six sieds, and then thaer ar all sorts of boxes and uther objekts which hav six sieds (eeven tho thaer mae not aulwaes be th saem length). Fienaly, if Jony gets a peess of caek or a sliess of peetsa, chanses ar guud too that it has bin cut up six waes, whether at hoem or at th baekery—sinss when was th last tiem that U saw a sweet or a  paestry sliest up into fifths!, probably never!! And if Jony stil isn’t satisfied, all he has to do is luuk at th ruum he hapens to be in; in adishon to th for wauls around him he shuud aulso noetis a seeling abuv him and a flor beneeth him! This meens, then, that th tipikal kid of his aej spends moest of his indor existanss surounded bi six surfases!!

Sinss we hav now demonstraeted that th number six is a qiet important and uesful number, we wuud agen expekt that it shuud be eezy to count bi sixes as wel. Unforchunatly, heer agen, poor Jony is going to be disapointed bi th desimal Hindoo-Aerabik sistem of noomeraeshon. If he tries counting bi sixes he gets th foloeing set of numbers:

 0, 6, 12, 18, 24, 30, 36…

It’s troo, that if he counts far enuf sum kiend of a repeeting patern duz emurj in th last dijit, but oenly after it has siekld throo every posibl eeven number, and eeven then thaer’s no lojikal patern to it sinss th wun-dijit endings apeer to jump up and doun in caeotik fashon. Thus, Jony discuvers that counting bi sixes is no fun at allL. Mor than that, Jony has discuverd in efekt that baess ten is not uesful for efishently counting enything in th reeal wurld that eether needs to be counted or can be counted!!

In akchuality, however, Jony probably woen’t eeven bother trieing to count bi sixes to begin with; he gaev up long befor after he encounterd th aukwardnes of counting bi threes and fors. Eeven wurss, Jony, if he is liek moest tipikal kids his aej, woen’t eeven think of th posibility that counting bi threes, fors, or sixes cuud in theeory be eezy, let aloen that it aut to be. Bi th tiem Jony has reecht th second or thurd graed his expoezher to th od praktis of counting bi tens wil hav forever bliended him to th troo naecher of counting, and of noomeerikal reality in jeneral. Chanses ar, unles he becums a profeshonal mathematishan wun dae (or a gifted amachuur)  he wil never reealiez that thaer is sum uther method of counting which is much beter sooted to deeling with ordinaery dae to dae counting tasks, not to menshon th mor sofisticaeted consepts and aplicaeshons of arithmetik and hieer mathematiks. Th “number bliendnes” which he aqiers in urly graed skool wil advursly afekt him for th rest of his lief, with th rezult that he wil tipikaly gro up with th impreshon that mathematiks is an obscuer disiplin that has litl to do with uther sfeers of reality. It is no wunder, then, that when Jony gets oelder he wil probably tel his budys in hie skool and colej that “he haets math”, and that if oenly he cuud get past th mathematikal reqierments that he needs to get his particuelar degree he wuud gladly do so…so that he can get out of colej as qikly as posibl and get into th “reeal wurld” of wurk!!!

Th solooshon

At this point wun miet seeriusly wunder whether enything cuud akchualy be dun to aulter or mitigaet this trajik staet of afaers, or whether Jony is just faest with sum inexoribl “grim fakt of mathematikal naecher” that can not be chaenjd at all, eny mor than we can maek mathematiks “simpl” be “reguelariezing” th value of pi or th value of th sqaer root of too. Forchunatly, however, it turns out that thaer is a verry eezy solooshon to all of this, and it is a solooshon for which we aulredy noe th anser, and that anser is constantly hinted to us every dae as soon as we luuk at a clok faess or consult a calendar, or hapen to open a box of doenuts for th morning cofy braek at th ofis. Th solooshon, odly enuf, is that we hav to ad too mising noomerals to our tradishonal sistem of Hindoo Aerabik noomeraeshon! That is all thaer is to it! It turns out that th tradishonal habit of counting bi groops of ten is defektiv sinss naecher raerly if every groops objekts bi tens or fievs. Bi extending our “counting groop” bak to th fuul duzen, however, our sistem of noomeraeshon wil be braut bak into harmony with comonsenss prinsipls of counting, with th wae numbers akchualy wurk in th reeal wurldSJ

But, U mae ask, how is this posibl? How wuud this be dun in praktis? How wuud U stil hav plaess value if U uez th duzen instead of ten? Th best wae to demonstraet how this wurks is to sho how Jony shuud hav akchualy bin taut to count bi prezenting th modified “number lien” below:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, E, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1T, 1E, 20, 21…

In this exampl heer U wil imeediatly noetis that too nue “numbers” hav bin aded between 9 and 10!! Th capital leters T and E stand for “ten” and “eleven” respektivly. But whot about th number that is laebld as ‘10’?: This number can now be refurd to as “a duzen” or “twelv”, whichever wun prefers. It reealy duzn’t mater in this caess. Thaer ar now no longger eny “teens” such as thurteen, forteen, fivteen, and so on; after a duzen U simply sae “duzen-wun, duzen-too, duzen-three…” and so on. Thus when Jony counts in this nacheral wae of counting it goes as foloes:

Zeero, wun, too, three, for, fiev, six, seven, aet, nien, ten, eleven, duzen, duzen-wun, duzen-too, duzen-three, duzen-for, duzen-fiev, duzen-six, duzen-seven, duzen-aet, duzen-nien, duzen-ten, duzen-eleven, too duzen, too duzen-wun, and so on.

At this point wun can imeediatly see that Jony is now aebl to count bi duzens enywhaer, whether counting boxes of doenuts, cartons of egs, too six-paks of his faevorit soft drink, th munths in th yeer, th ours in a haf dae, th number of inches in a tradishonal fuut, th number of “haf-steps” in a muezikal oktav, or simply for counting things in jeneral whether thae cum in duzens or not!.

But, wun mae ask, how duz this akchualy maek counting eny eezyer? We need to be paeshent, tho, becauz we ar just beginning to explor th uesfulnes of th nacheral sistem of counting. Now how wil things luuk it Jony tries counting bi th eeven numbers? In this caess we get th foloeing set of numbers:

 0, 2, 4, 6, 8, T, 10, 12, 14, 16, 18, 1T, 20, 22, 24, 26, 28, 2T, 30, 32, ets.

It turns out that when counting bi th eeven numbers thaer is a nue eeven number, T, that needs to be counted befor going on to 10. Thus Jony wil sae at this point, “…six, aet, ten, duzen, duzen-too, duzen-for…”  and so on. For th od numbers th patern wil luuk liek this:

 1, 3, 5, 7, 9, E, 11, 13, 15, 17, 19, 1E, 21, 23, 25, 27, 29, 2E, 31, ets.,

 and Jony wuud sae th seeqenss out loud as “…fiev, seven, nien, eleven, duzen-wun, duzen-three…” and so on. Thaefor, Jony has to deel with a nue eeven number, T, cauld ‘ten’, and a nue od number, E, cauld ‘eleven’, and thees nue numbers must aulwaes be sed befor going on to th next hieer duzen, no mater how hie Jony counts.

But, U mae protest, duzn’t this akchualy maek counting mor dificult, sinss Jony now has to lurn eeven mor simbols, and th number naems to go along with them, just in order to count bi wuns or toos? As we wil soon see, however, this is a relativly smaul sacrifiess to maek when we taek into acount th incredibl benefits of being aebl to count in th nacheral maner, as opoezd to th artifishal sistem of counting on wun’s finggers and thums. Now let’s luuk at counting bi threes. This wuud be th next lojikal number to count bi, and Jony wil now discuver to his joy and satisfakshon that counting bi threes is just as simpl and eezy as counting bi wuns or toos—and in fakt, eeven eezyer! Jony’s atempt to count bi groops of three in th nacheral counting sistem goes as foloes:

0, 3, 6, 9, 10, 13, 16, 19, 20, 23, 26, 29, 30, 33, 36, 39, 40, 43, ets.

Now, whot’s hapend heer? This tiem thaer ar no nue simbols or number naems for Jony to lurn, but luuk at th reguelar patern of th numbers!! From our starting point at zeero, we start off as we wuud with th tradishonal desimal sistem, counting “three, six, nien…”, but then we ar bak at a duzen agen, and a duzen ends in zeero, so we continue on as befor, counting “duzen-three, duzen-six, duzen-nien, too duzen, too duzen-three…” and so on. But this is miraculus! All multipls of three aulwaes end in eether -0, -3, -6, or -9, and this siekl continues on forever, no mater how hie we count! Now Jony is so exieted with his discuvery that he can hardly waet to get hoem from skool to tel his muther and faather whot he lurnd in skool that dae! Moroever, it turns out that not oenly do all multipls of three end in eether zeero, three, or nien, but a number is divizibl bi three oenly if it ends in eether zeero, three, or nien! Numbers that end in eny uther number, such as -1, -2, -4, -5, -7, -8, -T, or –E, ar never eevenly divizibl bi three, not under eny surcumstanses!! Jony is thus not oenly lurning how to count efishently but he is lurning about multiplicaeshon, divizhon, and divizibilty at th saem tiem! In fakt, it is now just as eezy to tel whether a number is a multipl of three or not as it is to tel whether a number is eeven or od. This discuvery then leeds to an interesting inoevaeshon: In analojy to th wurd “eeven” for multipls of too, it is now posibl to coin th turm “threeven” for multipls of three! Thus thoes numbers which end in 0, 3, 6, or, 9 ar threeven, and all uther numbers ar “throd”.

Jony’s advencher in arithmetik is oenly just begining, however. Whot hapens if he venchers to tri counting bi fors? Now, as U wil recaul urlyer, Jony did not hav a verry pozitiv expeeri’enss when trieing to count bi fors in th artifishal desimal sistem, but let’s see how he faers when he uezes nacheral counting bi twelvs. This tiem th seerys he gets is as foloes:

0, 4, 8, 10, 14, 18, 20, 24, 28, 30, 34, 38, 40, 44, 48, 50, 54, ets.

Boy, this is eeven eezyer than counting bi threes! This tiem it is oenly nesesaery to remember that a number can oenly be a multipl of 4 if it ends in 0, 4, or 8. It’s hard to get much simpler than that! Liekwiez, sinss it is now so simpl to tel whether a number is divizibl bi for we can cum up with a speshal turm for such numbers, and caul them “freeven”, from “for + eeven”. Numbers which ar not divizibl bi for, then, can simply be cauld “frod”, from “for + od”.

At this junkcher, howeve, sumwun is bound to ask Jony about counting bi fievs. Wel, whot about counting bi fievs? As U recaul from Jony’s expeeri’enss urlyer, counting bi fievs was verry eezy in th ten-finggerd and thumbd desimal sistem, but thaer was verry litl point in it. Verry fue things akchualy cum in fievs, whether we ar tauking about groops of objekts, counting th sieds of objekts in eether too or three dimenshons, or divieding up objekts into peeses or slieses. Sinss it is not verry important to be aebl to noe whether a given number is a multipl of fiev or is eevenly divizibl bi fiev we can saefly skip this topik for th tiem being and get on to th ishoo of counting bi th next larjer number, six.

Now, sinss six is boeth an eeven number and a multipl of three (in uther wurds, an “eeven threeven”) we wil reech th number six regardles of  whether we ar counting bi wuns, toos, or threes, so Jony is aulredy qiet familyar with th number six at this point. In fakt, it is qiet eezy to figyer out how to count bi sixes if U simply count bi threes and then eliminaet all thoes multipls of three which ar od. U then get a seerys of numbers which progreses as foloes:

0, 6, 10, 16, 20, 26, 30, 36, 40, 46, 50, 56, 60, 66, 70, 76, 80, 86, ets.

This is just purfekt! This is th wae it is supoest to be. Now U noe whi thae sae, “six of wun, haf a duzen of anuther”! If this patern aulredy luuks familyar, it is becauz it is th saem patern that U get if U count bi six munths at a tiem, with th duzens heer being eqivalent to yeers, or if U count bi six inches at a tiem, and then count in feet and inches, or if U count duzens and haf-duzens of doenuts.

Thus U can see now that when Jony counts th nacheral wae in turms of duzens, whether he counts bi wuns, toos, threes, fors, or sixes th prosess of counting turns out to be verry simpl, as U can see from th foloeing compaerison:

Bi wuns:   0,   1,   2,   3,   4,   5,   6,   7,   8,   9,   T,   E, 10, 11, 12, 13, 14, 15, 16, 17

Bi toos:     0,   2,   4,   6,   8,   T, 10, 12, 14, 16, 18, 1T, 20, 22, 24, 26, 28, 2T, 30, 32

Bi threes:  0,   3,   6,   9, 10, 13. 16, 19, 20, 23, 26, 29, 30, 33, 36, 39, 40,  43, 46, 49

Bi fors:     0,   4,   8, 10, 14, 18, 20, 24, 28, 30, 34, 38, 40, 44, 48, 50, 54,  58, 60, 64

Bi sixes:    0,   6, 10, 16, 20, 26, 30, 36, 40, 46, 50, 56, 60, 66, 70, 76, 80,  86, 90, 96

This is whot nacheral counting is all about. Now Jony no longger haets math!

 Moroever, as U wil see in fuecher relaeted artikls on this subjekt, within a fue yeers Jony wil verry liekly get a fuul too yeers ahed of his aej groop in math compaerd to thoes of his aejmaets he ar stil uezing th antiqaeted desimal sistem. Cuud U imajin whot it wuud meen if all skool children wer a fuul too graed levels ahed in turms of thaer mastery of arithmetik consepts? Now do U begin to see how important this ishoo is for th fuecher of ejoocaeshon?....

(Deer frend of troo math: This is th end of th furst sekshon of this prezentaeshon of th amaezing benefits of baess twelv and th nacheral sistem of counting. This is just a verry baesik introdukshon to th sistem, and U no dout stil hav a lot of qeschons about it that hav not bin anserd. In th weeks and munths to cum thaer wil be mor instaulments in this lekchen seerys which wil fil in th rest of th informaeshon that U need to noe. In th meen tiem, however, U hav aulredy embarkt on th furst steps tord liberaeting yuur miend from th curss of number-bliendnes and desimal “thaut controel”;-). Mor seeriusly, however, bi shoeing an interest in lurning mor about th nacheral sistem of duzenal counting and of arithmetik U ar demonstraeting to yuurself and to uthers around U that U ar comited to th fuulist uess of reezon and lojik to as meens of solving problems, and that U ar not alowing yuurself to be dominaeted bi arbitraery customs and convenshons, espeshaly if thees can be shoen to no longger adeqotly surv th funkshons thae had bin orijinaly intended to surv in sosieety. U ar thus demonstraeting th spirit of free-thinking, oepen-miendednes, and a flexibl aproech to problem solving that wil maek U a valueabl aset to eny employer and which wil get U far ahed in lief, no mater whot kiends of career plans U hav. Ultimatly, however, th batl for a rashonal arithmetik sistem is a part of th larjer strugl agenst all ilojikal and arbitraery sistems in sosieety, sinss an inefektiv or inefishent sistem is ultimatly, in th long run, an unjust sistem, and th fiet agenst all unjust sistems aulwaes starts with a singgl step. U hav just taeken that step!)